Mr. H. S. Warner on Conjugate Points, 85 



A C of the triangle ABC, that is, it will be an infinite straight 

 line. Even if the imaginary quantity were considered as a 

 real quantity, measured along a line perpendicular to the 

 plane of the co-ordinates x and y, which is what is meant, I 

 suppose, by " the imaginary curve being viewed as real curve 

 out of the plane of co-ordinates," the expression 0^-1 

 would indicate a point in, and therefore assignable on, that 

 plane, for it would indicate a point neither above nor below, 

 and therefore in, it. 



3. Similarly also, the hyperbola will become in the ulti- 

 mate state an infinite straight line, and not, as Prof. Young 

 says, " two infinite straight lines in directum with the axis of 

 abscissas, but separated from each other by an interval equal to 

 the fixed principal diameter." Having, I think, made it evi- 

 dent that it is an error to consider 0\/-~l as unassignable, it 

 being really equivalent to 0, I shall now consider the theory- 

 of conjugate points. 



4. If y = <p (x) be the equation to a curve, and we can put 

 <p (x) under the form./! [x) +/ 2 (x), so that 3/=^ {x)+f% (x), 

 and if we lay down the curve which is the locus of y —f x (x), 

 then to determine the locus ofy= $ (x), for any value of x, we 

 shall merely have to measure along the ordinate drawn at that 

 value of x, a distance equal to the value ot'f 2 (x) corresponding 

 to the given value of x ; measuring from the intersection of 

 the ordinate and the curve y =/\ (x) ; this curve we may call 

 the axial curve, the other one, y = <J> (<r), being derived by mea- 

 suring from it as from an axis. If at any point of the axial 

 curve^ (x) is = 0, then that point is a point in the locus of 

 y = <p(x) also; this is evident. Also, if for any value of x, 

 f^ (x) is impossible, then at that value of x the curve y =4> (x) 

 will be unassignable, for we cannot assign any distance to be 

 measured along the ordinate, and hence cannot exhibit fhe 

 curve. 



5. If now, for a continuous series of values of x i Jl l (x) is of 

 the form A + B \/ — 1, the curve will be unassignable for all 

 these values of x (by No. 4), forJ^(x) will be impossible; but 

 if continuing the series, we at last reach a value of x at which 

 B = 0, so thatj^ (x) is of the form A+ Ov — 1, the curve will 

 be assignable, for A + */— 1 =A + = A (by No. 1); and if 

 further A = 0, it will be a point of the axial curve correspond- 

 ing to the same value of x, that will be the locus. If now, 

 still continuing the series of values, B continues = 0, orj^ (x) 

 ceases to be impossible, the imaginary quantity */•— 1 disap- 

 pearing, the locus will be assignable; and thus we shall have 

 a real curve commencing at that ordinate at which B = 0. 

 But if, on the other hand, on continuing the series, the ima- 



